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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2405.02557 |
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| _version_ | 1866929335455186944 |
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| author | Bae, Junsik Kim, Yunjoo Kwon, Bongsuk |
| author_facet | Bae, Junsik Kim, Yunjoo Kwon, Bongsuk |
| contents | We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for the blow-up and the temporal blow-up rate along the characteristic curve, we rather give a constructive proof of singularity formation from smooth initial data. More specifically, employing the stable blow-up profile of the Burgers equation in the self-similar variables, we establish the global stability estimate in the self-similar time, which yields the asymptotic behavior of blow-up solutions near the singularity point. Our analysis indicates that the smooth solution to the Euler-Poisson system can develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time, while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial data are sufficiently close to the blow-up profile in some weighted $C^4$-topology. We also present a similar result for the isentropic case, and discuss noteworthy differences in the analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure of singularities for the Euler-Poisson system of ion dynamics Bae, Junsik Kim, Yunjoo Kwon, Bongsuk Analysis of PDEs We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for the blow-up and the temporal blow-up rate along the characteristic curve, we rather give a constructive proof of singularity formation from smooth initial data. More specifically, employing the stable blow-up profile of the Burgers equation in the self-similar variables, we establish the global stability estimate in the self-similar time, which yields the asymptotic behavior of blow-up solutions near the singularity point. Our analysis indicates that the smooth solution to the Euler-Poisson system can develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time, while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial data are sufficiently close to the blow-up profile in some weighted $C^4$-topology. We also present a similar result for the isentropic case, and discuss noteworthy differences in the analysis. |
| title | Structure of singularities for the Euler-Poisson system of ion dynamics |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.02557 |